I don't own a factory rifle today, but if I could go back in time, I would have broken in any factory barrel that I owned.
Todays custom barrels are far better than they were 7-8 yrs ago, as far as staying clean, they are hand lapped so well. Still, I clean after a simple velocity check, 10-20 rds. but I scope the barrel first, if I see excessive copper in the first part of the barrel, I clean with an abrasive, JB bore paste or Witches Brew and try iron out any noticeable rough spots. If the barrel is just dirty, I just use a carbon solvent for a quick clean up. I will have a load in the next 25 shots, after which I will clean again. Then load enough to get the barrel through it's speedup, if the load holds, I will take my rd ct on the barrel up to around 150rds and clean one more time. Then I will fire it 2-300 times to determine how long I can go between cleanings with it.
Let me say too, I always jag a new barrel with Kroil to get any remaining cutting fluids out before I fire the first shot.
As for cooling a barrel, I would not blow anything down my tube except ambient temp air. I carry a riflekuhl barrel cooler, small, handy, though a battery eater, and they are not cheap, but gets the job done. Even though I have a designated range vehicle, I am not going to need a cart to haul my crap to the firing line, but an air mattress pump and correct size tubing would be the ticket. If you are waiting on a barrel to cool, move some air through it.
Lol, I doubt any 2 of us do the same thing here.